As published Malasics BBA 2009

Author's personal copy Biochimica et Biophysica Acta 1788 (2009) 2471–2480

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Biochimica et Biophysica Acta j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / b b a m e m

Protein structure and ionic selectivity in calcium channels: Selectivity ﬁlter size, not shape, matters Attila Malasics a, Dirk Gillespie b, Wolfgang Nonner c, Douglas Henderson d, Bob Eisenberg b, Dezső Boda a,d,⁎ a

Department of Physical Chemistry, University of Pannonia, P. O. Box 158, H-8201 Veszprém, Hungary Department of Molecular Biophysics and Physiology, Rush University Medical Center, Chicago, IL 60612, USA Department of Physiology and Biophysics, Miller School of Medicine, University of Miami, Miami, FL 33101, USA d Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA b c

a r t i c l e

i n f o

Article history: Received 5 March 2009 Received in revised form 28 September 2009 Accepted 30 September 2009 Available online 7 October 2009 Keywords: Calcium channel Selectivity Permeation Grand canonical Monte Carlo simulation

a b s t r a c t Calcium channels have highly charged selectivity ﬁlters (4 COO− groups) that attract cations in to balance this charge and minimize free energy, forcing the cations (Na+ and Ca2+) to compete for space in the ﬁlter. A reduced model was developed to better understand the mechanism of ion selectivity in calcium channels. The charge/space competition (CSC) mechanism implies that Ca2+ is more efﬁcient in balancing the charge of the ﬁlter because it provides twice the charge as Na+ while occupying the same space. The CSC mechanism further implies that the main determinant of Ca2+ versus Na+ selectivity is the density of charged particles in the selectivity ﬁlter, i.e., the volume of the ﬁlter (after ﬁxing the number of charged groups in the ﬁlter). In this paper we test this hypothesis by changing ﬁlter length and/or radius (shape) of the cylindrical selectivity ﬁlter of our reduced model. We show that varying volume and shape together has substantially stronger effects than varying shape alone with volume ﬁxed. Our simulations show the importance of depletion zones of ions in determining channel conductance calculated with the integrated Nernst–Planck equation. We show that conﬁning the protein side chains with soft or hard walls does not inﬂuence selectivity. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Ion channels control many biological functions from signalling in the nervous system to initiating muscle contraction. These channel proteins form pores that conduct ions down their electrochemical potential gradient across otherwise impermeable membranes of both the cell and the organelles within it. Ion channels can select which ionic species they conduct, with different channel proteins conducting different ionic species. For example, potassium channels conduct K+ and (statistically speaking) exclude Na+. The quite different sodium channel protein conducts Na+ and excludes K+ [1]. Another important class of ion channels preferentially conducts Ca2+ compared to monovalent cations like Na+ and K+. A biologically important subclass of calcium channels are highly Ca2+ selective. For example, the L-type calcium channel has a Ca2+ afﬁnity of 1 μM; in 30 mM Na+ as little as 1 μM Ca2+ reduces the Na+ current measured in 0 Ca2+ by 50% [2,3]. While the main physiological function of the L-type calcium channel is to conduct Ca2+ in the background of a much higher concentration of monovalent cations, it also discriminates between ions of the same charge (e.g., Ca2+ vs. Ba2+ or Na+ vs. K+ [4]). ⁎ Corresponding author. Department of Physical Chemistry, University of Pannonia, P. O. Box 158, H-8201 Veszprém, Hungary. E-mail address: [email protected] (D. Boda). 0005-2736/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.bbamem.2009.09.022

The physical mechanism of the selectivity of the L-type calcium channel has often been discussed because of the profound medical importance of the channel. In the ﬁrst theories of permeation, ions were said to “hop” over energy barriers (reviewed by Sather and McCleskey [5], criticized by Eisenberg et al. [6–9]). This kind of helpful but phenomenological description has been replaced by modern theories and computer simulations. Corry et al. [10–12] have studied Ca2+ selectivity, but their simulations, using canonical ensembles, necessarily used Ca2+ bath concentrations N18 mM — 10,000 times larger than the micromolar afﬁnity of the pore. When grand canonical Monte Carlo (MC) simulations were performed using the model of Corry et al. [13] for the L-type calcium channel, they showed only 200 μM Ca2+ afﬁnity, much less than the natural channel. Just as signiﬁcantly, their model cannot distinguish between ions of the same valence, but different size [11,12] even though natural channels can do this. So far, the theory that best describes experimental properties of calcium channels is the charge/space competition (CSC) mechanism. This theory not only produces micromolar Ca2+ afﬁnity for the L-type calcium channel [14–16] but also describes how different calcium channel subclasses can have the wide range of Ca2+ afﬁnities seen in experiments [14]. The CSC model correctly distinguishes between monovalent cations of different size [16–21]. The model uses only a few adjustable parameters to account for the main selectivity

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properties of the calcium channel and its cousin, the sodium channel [22,23]. The CSC mechanism describes ion selectivity in calcium and sodium channels as a balance of the ions' electrostatic attraction into the channel's highly charged selectivity ﬁlter and the ions' excluded volume in the small selectivity ﬁlter that is crowded with ions and protein side chains [14–31]. In this mechanism, selectivity of Ca2+ over monovalent cations increases when the number of cations in the pore increases (with the shape and size of the pore unchanged). It also increases when the radius of the pore decreases. Thus, pore volume is a critical parameter for Ca2+ versus monovalent cation selectivity [24]. Another important parameter is the dielectric coefﬁcient of the protein that surrounds the permeation pathway; a low protein dielectric coefﬁcient (relative to the pore) attracts more cations into the pore and thereby increases Ca2+ afﬁnity [14,19]. The role of the pore volume in selectivity was ﬁrst suggested by Nonner et al. [24]. In fact, they suggested that pore volume is the fundamental determinant of Ca2+ versus monovalent cation selectivity. Both theoretical [14,17] and experimental [30,31] studies have shown that reducing the pore radius has large effects on this kind of selectivity. To test the CSC mechanism experimentally, Miedema et al. [30,31] chemically modiﬁed the pore of the otherwise non-selective OmpF channel. They showed that adding negative charge turned the non-selective porin into a calcium channel and reducing the pore volume improved Ca2+ selectivity. In this paper, we test the “volume-is-fundamental hypothesis” directly by changing the shape of our model calcium channel. By “shape” we mean the pair of H and R parameters of our reduced model, namely, the length and radius of the cylindrical selectivity ﬁlter. We change these two parameters in two ways: (1) by simultaneously changing the volume varying R at ﬁxed H (making the pore wider/narrower) or varying H at ﬁxed R (making the pore longer/shorter) and (2) by keeping the volume ﬁxed (changing the pore from a long and narrow to a short and wide cylinder). Reducing the pore length (while increasing the pore radius) changes packing effects of the hard-sphere ions both axially and radially. In complex systems like ion binding proteins, details of structure are believed to be of the greatest importance in determining binding selectivity (i.e., how many ions accumulate in the pore) and so one would expect channels of different length to have different selectivity, even if they have the same volume. We ﬁnd, however, that changes in pore length have relatively little effect on Ca2+-binding afﬁnity if volume is held constant. If the volume is changed simultaneously with the H and R parameters, however, selectivity can change substantially in a wide range. Stronger packing in the axial dimension does have effects on ion currents. It produces deeper depletion zones (zones where the density of certain type of ions is signiﬁcantly small). It was clear earlier [32] and it was shown explicitly recently [15,21] that these depletion zones determine the amount of current each ionic species conducts, similar to the role of depletion zones in transistors [7,33]. In this paper we analyze the effect of these depletion zones in terms of the integrated Nernst–Planck equation for various shapes of the channel. The effect of the nature of conﬁnement of protein side chains is also examined and found to have small effects.

of calcium channels have not yet been reported. This large-scale structure of alpha helices and the backbones provides a pore with a relatively rigid wall as implied by the stable currents ﬂowing through these channels [37,38]. At the same time, channel proteins have a certain amount of ﬂexibility because they are in a constant thermal motion at room temperature. The most ﬂexible parts of the protein are the side chains. To take into account both aspects in our reduced model, we use a rigid conﬁnement modeled by a pore with hard walls as shown in Fig. 1 and we place mobile protein side chains (the carboxyl groups) inside this conﬁnement. The region −H/2 b x b H/2 deﬁnes the selectivity ﬁlter. The two important geometrical parameters of this model are the radius R and the length H of the ﬁlter. Fig. 2 shows snapshots of three different channel shapes determined by three different sets of the R and H values. The CSC model states that the packing fraction (the volume occupied by the bodies of the side chains relative to the ﬁlter volume) and the charge of these side chains are the main determinants of selectivity. Therefore, we chose the simplest model that includes both the charge of the side chains and the volume that they occupy: each carboxyl group is represented as two half-charged, independent oxygen ions (two O1/2−) that are free to move within the selectivity ﬁlter but are prevented from leaving it (illustrated by red spheres in Fig. 2). The diameter of the oxygen ions is dO = 2.8 Å. The oxygen ions are conﬁned within the ﬁlter by a soft conﬁning potential that is deﬁned by the following Boltzmann factor expð− Uconf ðxÞ = kT Þ = 0:5 + 0:5 tanh ½Faðx F Hc Þ;

ð1Þ

where Hc = H/2 − dO/2, x is the longitudinal coordinate along the axis of the channel, and the signs are positive for x b 0 and negative for x N 0. Fig. 3 shows this function for three values of the a parameter that deﬁnes the “softness” of the wall. In the limit a → ∞ the usual hard wall conﬁnement is recovered. The Boltzmann factor describes the

2. Model and methods 2.1. Simple model of a calcium channel The selectivity ﬁlter of the L-type calcium channel contains four glutamates (E) that determine the channel's selectivity properties [34,35] (the EEEE locus). Each glutamate has a terminal COO− group so the pore has a charge of − 4e. Channel proteins have an atomic structure that in favorable cases can be determined from X-ray diffraction experiments, usually at 100 K [36], although the structures

Fig. 1. The cross-section of the channel model used in our simulations. The threedimensional model is obtained by rotating it around the x-axis. The two important parameters – changed in this paper – are the length H and radius R of the selectivity ﬁlter. The ﬁgure focuses on the channel and does not show the entire baths; the size of the simulation box is much larger than indicated in the ﬁgure. Distances are measured in Ångström throughout this paper.

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charge) and (2) enhances electrostatic attraction between oppositely charged ions (through the interaction with the induced charge of the counter-ions). These effects are thought to be present in real channels. The second term dominates in the highly charged selectivity ﬁlter of calcium channels and makes the ﬁlter more selective for Ca2+ over Na+ as ɛpr is decreased [29]. The case when the pore has a dielectric coefﬁcient that is lower than ɛw (due to, for example, dielectric saturation [39]) is computationally problematic because of ions crossing sharp dielectric boundaries. The Na+, Ca2+ and Cl− ions are free to move everywhere in the solution (in the channel and in the baths). They are modeled as charged hard spheres with diameters 1.9, 1.98 and 3.62 Å, respectively. The simulation cell is a large ﬁnite cylinder that includes the baths, the membrane and the channel protein. The simulation cell is chosen to be large enough so that (1) bathing solutions have bulk properties independent of special effects at the walls of the simulation cell and (2) results of simulations do not change as the simulation cell is increased in size. This same model of the EEEE locus has been used to replicate experimental data of the L-type channel and other calcium channels with success [14,15,17–21,24–29]. A nearly identical model accounts for a wide range of data from the DEKA sodium channel and its DEEA mutant (which behaves like a calcium channel [40]) with the same two parameter values, despite the very different properties of the two channel types [23]. A closely related model accounts for a wide range of data for the RyR calcium channel and in fact provided detailed predictions of experimental results before they were measured [18–21]. 2.2. Monte Carlo simulations Details of our Metropolis MC simulations are given elsewhere [13,29,41,42]. The ionic concentrations in the bath have to be established accurately because the main output of calculations is the ionic content of the channel as a function of bath electrolyte composition. In order to efﬁciently simulate micromolar concentrations for Ca2+ in the bath, we used the grand canonical ensemble [43], where the simulation cell is connected to an external bath of ﬁxed chemical potentials through insertion/deletion of neutral ionic groups (1 Na+ + 1 Cl− or 1 Ca2+ + 2 Cl−). The chemical potentials that correspond to prescribed concentrations of these salts were calculated with an iterative grand canonical MC method of Malasics et al. [44]. To accelerate convergence, special insertion/deletion steps were applied, where Na+ or Ca2+ were inserted (or deleted) into (or from) the channel itself instead of being inserted (or deleted) into (or from) the entire simulation cylinder [13].

Fig. 2. Panels A–C show snapshots of the packing of the 8 O1/2− ions (red spheres) in the pore axially. The three panels show snapshots of three different geometries used to study the effect of pore shape in Section 2. Green and blue spheres represent Ca2+ and Na+, respectively.

probability that an oxygen ion can be found somewhere along the xaxis. Its integral Veff = A∫exp(−Uconf (x) / kT)dx can be identiﬁed as the effective volume that is available for the oxygen ion and is the same for different values of a: Veff = 2AHc. The dielectric coefﬁcient of the electrolyte solution is ɛw = 80, while the dielectric coefﬁcient of the protein is ɛpr = 10. The low dielectric environment of the pore (1) creates a dielectric barrier for the passing ion (through the interaction with its own induced

Fig. 3. The Boltzmann factor that conﬁnes oxygen ions in the selectivity ﬁlter for various values of parameter a that tunes the “softness” of the conﬁning potential Uconf (x) (Eq. 1).

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The heart of the MC simulation is the energy calculation. In the case of this model, the system contains dielectric boundaries where a polarization surface charge is induced by the ions. This induced charge is determined in every simulation step by the induced charge computation (ICC) method [45], a numerical method to calculate the discretized surface charge on the dielectric interfaces. The method was thoroughly tested and has been shown to be efﬁcient and accurate [24,29,45,46]. The main results of the simulations are the axial density proﬁles ci(x) of ionic species i. They are calculated as the average number of a given ionic species in predeﬁned slabs along the x-axis divided by the volume of the slab that is accessible to the centers of the ions. 2.3. The integrated Nernst–Planck equation The MC method provides equilibrium density proﬁles of the various ionic species. In recent papers [15,16], we have used an integrated form of the Nernst–Planck equation to relate equilibrium MC proﬁles to current. The Nernst–Planck equation (also known as the drift-diffusion or electrodiffusion equation) is −Ji ðrÞ =

1 D ðrÞci ðrÞjμ i ðrÞ; kT i

ð2Þ

where Ji(r) is the particle ﬂux density of species i, k is the Boltzmann's constant, T is the temperature, Di(r) is the possibly positiondependent diffusion coefﬁcient, and μi(r) is the chemical potential. This chemical potential includes the ideal term (kT log ci(r)) and an excess term due to correlation between hard sphere ions and the external voltage across the membrane V. We assume that (1) the resistance of the selectivity ﬁlter limits the ﬂux, (2) the chemical potential is uniform in the radial dimension, (3) the diffusion coefﬁcients are constant in the selectivity ﬁlter, (4) the two baths are identical, but at different constant electric potentials, and that (5) the applied voltage V is small so that the current/voltage relation is linear. With these approximations, the conductance of the channel for ionic species i is

gi =

z2i e2 Di 1 : Z kT 1 H = 2 dx A −H = 2 ci ðxÞ

ð3Þ

and the total conductance is g = Σi gi. Stevens [47] presents this equation, but it seems not to have been used in subsequent literature. Eisenberg [48] shows that the conductance is a rate constant equal exactly to the conditional probability of a well-deﬁned stochastic process. This equation shows why depletion (low concentration) zones can be so important. The conductance depends on the reciprocal of the integral of the reciprocal of the density proﬁle. Neighboring slabs of the channel along the ionic pathway can be represented as resistors connected in series [15,16]. If the number of charge carriers (ions) in any of these slabs is depleted (ci(x) is close to zero), then the integral for this slab (the resistance of the slab) becomes large making the whole resistance large. Its reciprocal (conductance) then becomes small. Thus, deep depletion zones for a given ionic species make the conductance of the channel for this ionic species small as discussed at length by Nonner et al. [32]. They were certainly not the ﬁrst to understand the importance of depletion zones. Depletion zones determine most properties of semiconductor devices like transistors [33,49,50]. In current/voltage measurements for ion channels, currents (conductances) are commonly given as normalized quantities. In particular, for the classic Ca2+-block experiment [2,3], where CaCl2 is added to a ﬁxed NaCl background (30 mM, in this work), the current (conductance) at a given [CaCl2] is normalized by the Na+ current

(conductance) in the absence of Ca2+. Thus, the total normalized conductance of the channel is Z

Z dx dx 2 zCa DCa c0Na ðxÞ c0Na ðxÞ Z + 2 ; g4 = Z dx dx zNa DNa cNa ðxÞ cCa ðxÞ

ð4Þ

where the superscript 0 refers to the density proﬁle at zero [CaCl2], while cNa(x), cCa(x) and g⁎ are taken at a given [CaCl2]. The ﬁrst term of the sum is the normalized Na+ conductance, while the second term is the normalized Ca2+ conductance. The density proﬁles are obtained from simulations and the only adjustable parameter of this model is the ratio of the diffusion coefﬁcients of Ca2+ and Na+ in the ﬁlter. For this, the value DCa / DNa = 0.1 was used (the mobility of Ca2+ is much smaller in the channel, we suppose, because of the large electrostatic attraction imposed by the oxygen ions). The CaCl2 concentration at which Ca2+ block occurs is insensitive to this parameter [15]. 3. Results and discussion Our basic setup is that of the classical Almers et al. [2,3] experiment which added CaCl2 to a background concentration of 30 mM NaCl using channel parameters in our model of R = 3.5 Å, H = 10 Å and ɛpr = 10. These reproduce the micromolar Ca2+ block. We change the ﬁlter radius R and ﬁlter length H in various ways to study the effect of these parameters on selectivity. In the experimental literature, selectivity is commonly characterized by [CaCl2]1/2, the [CaCl2] at which the current through the channel is half of that found when [Ca2+] = 0, i.e., g⁎ = 0.5 (from Eq. 4). The smaller [CaCl2]1/2, the more selective the pore is for Ca2+. To summarize the dependence of selectivity on various combinations of the parameters H and R, in Fig. 4 we plot [CaCl2]1/2 as a function of the ﬁlter volume (π R2 H). How steeply [CaCl2]1/2 changes with volume indicates how sensitive the Ca2+ afﬁnity is to ﬁlter volume. On the other hand, for a ﬁxed volume, the ﬁgure shows how sensitive the Ca2+ afﬁnity is to ﬁlter shape (i.e., different combinations of H and R that keep the volume ﬁxed). Visually, the horizontal change indicates volume sensitivity, while the vertical change indicates shape sensitivity. The inset shows the same data in a logarithmic scale. A detailed discussion of the results in the ﬁgure is given in the text below.

Fig. 4. Summary of the simulation results of this work in terms of the [CaCl2]1/2, the concentration of CaCl2 at which channel conductance drops to half its value in the absence of CaCl2. This value [CaCl2]1/2 is plotted against ﬁlter volume for various experiments when ﬁlter length, ﬁlter radius or ﬁlter volume are kept constant. The inset plots a part of the main graph for small volumes with a logarithmic scale of the ordinate. The numbers near symbols indicate the values of H and R.

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3.1. Changing pore shape and volume simultaneously has strong effect on selectivity The effect of pore radius was studied in our previous work [14] by examining the occupancy of the pore by various ions for [NaCl] = 100 mM at ﬁxed channel length H. The concentration of CaCl2 at which the average number of Na+ drops to half its value in the absence of Ca2+ depends sensitively on pore radius. (As Ca2+ is added to the bath, Na+ in the ﬁlter is gradually replaced by Ca2+.) Smaller pore volume makes the selectivity ﬁlter even more crowded, which makes it even harder for cations to ﬁnd space in the high-density ﬁlter. This competition favors Ca2+ because Ca2+ provides twice the charge to balance the negative charge of the oxygen in the ﬁlter than Na+ while occupying about the same space. This competition can be described in terms of a balance of energy and entropy. By absorbing Ca2+ instead of Na+, the ﬁlter reduces free energy. It decreases electrostatic energy without decreasing entropy too much. (Note that entropy can also be changed by tethering oxygen ions together as in real COO− groups. Such details will be built into the model in future studies.)

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Because we ﬁrst used the integrated Nernst–Planck equation [15] after the publication of the above results [14], we repeated the simulations for [NaCl]=30 mM and analyze the results in terms of conductance. Keeping H = 10 Å ﬁxed, we change pore radius and compute the normalized conductance of the channel using Eq. 4 with Dca / DNa = 0.1. The channel shape is changed in a way that the pore is made wider keeping its length ﬁxed (open circles with solid line in Fig. 4). The conclusions that we can draw are the same that were drawn in Ref. [14]: Ca2+ versus Na+ selectivity improves considerably with decreasing ﬁlter volume. Density proﬁles (not shown) have more structure in narrower ﬁlters, those with smaller R. Then, we changed pore length keeping the radius R = 3.5 Å ﬁxed. Fig. 5 shows the density proﬁles for length H = 8 Å (left panels) and H = 12 Å (right panels) for different Ca2+ concentrations. Proﬁles in the short channel have more structure; the depletion zones are more pronounced, and Ca2+ is more efﬁcient in squeezing Na+ out from the ﬁlter compared to the long channel. In the short channel, the Na+ proﬁles decrease almost to half of the zero-Ca2+ proﬁles (magenta open circles) at [CaCl2] = 10− 7 M, while this happens only at [CaCl2] = 10− 6 M in the case of the long channel. Furthermore, as Ca2+ gradually replaces Na+ in the selectivity ﬁlter, the depletion

Fig. 5. Axial distribution proﬁles for Ca2+ (top panels) and Na+ (bottom panels) for a short (H = 8 Å, left panels) and a long (H = 12 Å, right panels) selectivity ﬁlter at different Ca2+ concentrations (R = 3.5 Å and ɛpr = 10). The insets of the top panels show the oxygen proﬁles for [CaCl2] = 0 M. The proﬁles are symmetric about x = 0 because the simulation system is symmetric. Any deviation from symmetry is a result of statistical uncertainty of the simulation. The inset of the bottom-left panel magniﬁes the depletion zone region as indicated by the arrow. Concentrations are given in M.

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zones of Na+ become more pronounced in the shorter channel (see inset of Fig. 5, bottom-left). Fig. 6a shows that as Ca2+ is added to the bath, the current drops at lower Ca2+ concentration in the short channel than in the long channel. The shift of the curves along the abscissa shows the change in Ca2+ selectivity as H (and so the volume) is varied. Fig. 4 shows this shift in term of [CaCl2]1/2 (ﬁlled red squares with dashed line). We can change the ﬁlter volume in a narrower regime by changing H at ﬁxed R (as opposed to changing R at ﬁxed H). The variation of [CaCl2]1/2 as a function of ﬁlter volume is still large as seen in the inset where the results for small volumes are shown on a logarithmic scale. This inset shows that log10[CaCl2]1/2 changes linearly with pore volume. The inset also shows that Ca2+ afﬁnity is more sensitive to changes in pore length than in pore radius.

Eq. 4 makes it possible to separate the currents carried by Ca2+ and Na+. Fig. 6b shows these currents with DCa / DNa = 0.1. When Ca2+ concentration of the bath is less than 10− 4 M, only a very small current is carried by Ca2+ (Fig. 6b). Na+ then carries almost all of the total current (Fig. 6a,b). This Na+ current, however, is reduced by low micromolar concentrations of Ca2+, to the extent that Na+ in the pore is displaced by Ca2+ (Fig. 6c). It is a rare event that a Ca2+ ion enters the selectivity ﬁlter through the depletion zones, but once it is there, it stays there for a long time. Therefore, this ﬁrst Ca2+ ion tends to dwell in the pore but does not contribute signiﬁcant current. Na+ ions, on the contrary, can enter and leave the ﬁlter and carry current but only in the time when the Ca2+ ion is not there. A substantial Ca2+ current starts to ﬂow at millimolar Ca2+ concentrations, at which a second Ca2+ tends to occupy the pore (Fig. 6c). Our simulations measure the long-time average of this process, showing a reduced Na+ occupancy and current. Therefore, for small bath Ca2+ concentrations (where Ca2+ current is vanishingly small), the total conductance curves (Fig. 6a) reﬂect only the Na+ conductances (Fig. 6b), which, in turn, reﬂect the Na+ occupancies (Fig. 6c). This implies that, in this regime, the selective binding of Ca2+ to the pore is the ﬁrst-order determinant of the variation of conductance with [CaCl2]. This selective binding is sensitive to the volume of the pore. Depletion zones of Na+ have little role in this because the Na+ concentration changes in the depletion zones just as quickly (as a function of [CaCl2]) as it changes in the binding site (compare the bottom-left panel of Fig. 5 and its inset). The qualitative ﬁndings of Fig. 6a agree with experiments. In a highly selective calcium channel with a small selectivity ﬁlter Ca2+ block of current happens at smaller bath [CaCl2] and current has a deeper minimum as in the case of the L-type calcium channel [2,3]. In a less Ca2+-selective calcium channel with a larger selectivity ﬁlter, Ca2+ block of current occurs at larger bath [CaCl2] with a less pronounced minimum as in the case of the RyR calcium channel [21]. 3.2. Changing pore shape under conserved volume has small effect on selectivity

Fig. 6. The classical anomalous mole fraction experiment: (a) normalized conductances as functions of log10[CaCl2] in M added to a 30-mM NaCl background for channels with varying ﬁlter lengths at ﬁxed radius (R = 3.5 Å and ɛpr = 10). The × symbols show the experimental results of Almers et al. [2,3]. Their results are different from ours at high [CaCl2] because in the experiments Ca2+ was added only to one side of the pore (increasing Ca2+ driving force at high [CaCl2] [15]), while in our equilibrium simulations Ca2+ was added symmetrically. (b) Normalized ionic conductances of the pore for Na+ and Ca2+ as given by Eq. 4. (c) The average number of Na2+ and Ca2+ ions in the selectivity ﬁlter. In panels b and c, increasing curves refer to Ca2+, while decreasing curves refer to Na+.

Fig. 4 shows that the shape of the pore keeping the volume ﬁxed has a real, but small effect on selectivity. Consider the results for volume 460.6 Å for which there are two points: one with H = 10 Å and R = 3.83 Å (open circle) and another with H = 12 Å and R = 3.5 Å (ﬁlled square). The difference in [CaCl2]1/2 between these two points implies that shorter pore with larger radius has better Ca2+selectivity properties than a longer pore with smaller radius at the same volume. To explore this effect, we performed additional simulations on channels with different shapes, but the same volume as the previously studied geometry H = 10 Å and R = 3.5 Å. Fig. 2 shows these channels with dimensions H = 8 Å and R = 3.913 Å, H = 10 Å and R = 3.5 Å and H = 12 Å and R = 3.195 Å. All have the same volume 384.65 Å3. Fig. 7 shows the density proﬁles of different species for different Ca2+ concentrations and channel lengths. Each panel shows results for a different bath Ca2+ concentration: in Fig. 7a, [Ca2+] = 0 M; in Fig. 7b, [Ca2+] = 10− 7 M; in Fig. 7c, [Ca2+] = 10− 6 M; in Fig. 7d, [Ca2+] = 10− 5 M. Each curve in a given panel is computed for a different pore length. The densities are multiplied by R2, so the height of curves are proportional to the average number of ions in the cross-section of the pore instead of the local density (compare to Eq. 3 for the conductance). Ca2+ proﬁles are absent in the case of [Ca2+] = 0 M, so we show the curves for the oxygen ions in the top panel of Fig. 7a. They are quite independent of bath Ca2+ concentration. In the longer pores, additional peaks in O1/2− concentration are found because there is space for additional layers of oxygen ions in the axial dimension. The distributions for Na+ and Ca2+ are quite different for various pore lengths due to different

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Fig. 7. Axial distribution proﬁles for (a) [CaCl2] = 0 M, (b) [CaCl2] = 10− 7 M, (c) [CaCl2] = 10− 6 M and (d) [CaCl2] = 10− 5 M for channels of various shapes. Solid, dotted and dashed lines refer to selectivity ﬁlters of lengths H = 12, 10 and 8 Å, respectively. The top and bottom panels show Ca2+ and Na+ proﬁles, respectively. The top panel of panel a shows oxygen proﬁles (they are practically the same for every Ca2+ concentration). The proﬁles are multiplied by R2 thus obtaining a quantity that is related to the average number of ions along the x-axis and to the conductance deﬁned by the one-dimensional Eq. 3. This proﬁle provides a more straightforward comparison between pores of different cross-sections.

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packing, but they change together as Ca2+ concentration changes, which implies that selectivity should be similar in the three cases. Stronger packing results in considerable depletion zones for Na+ in the short pore at the locations of the peaks in O1/2− proﬁles. These depletion zones decrease the absolute value of the conductance of the channel. The normalized conductance, however, is less inﬂuenced because we normalize by the conductance at [CaCl2] = 0 M, where these depletion zones are also present. As pore length is increased, these depletion zones become less pronounced. Fig. 8 is analogous to Fig. 6. It shows the same quantities in panels a–c that Fig. 6 does as functions of log10[CaCl2], but here the pore radius is not constant. The curves in Fig. 8a, computed for selectivity ﬁlters with the same volume, are closer together than those in Fig. 6a, which were computed for selectivity ﬁlters with different volumes, showing that changing pore shape has a larger effect on selectivity if we also change the pore volume. Fig. 4 shows this result in terms of [CaCl2]1/2 (ﬁlled triangles with dotted line).

Fig. 9. Axial distribution proﬁles for O1/2−, Ca2+ and Na+ (from top to bottom) for various values of the “softness” parameter a for [CaCl2] = 10− 6 M (channel parameters are R = 3.5 Å, H = 10 Å and ɛpr = 10).

Similar conclusions can be drawn from Fig. 8 as from Fig. 6: The occupancy curves for Na+ (Fig. 8c) behave similarly to Na+conductance curves (Fig. 8b) in the regime of Ca2+ block ([CaCl]2 b10− 5 M), which implies that the block is a result of selective binding of Ca2+ over Na+ in the selectivity ﬁlter; that is, as the number of Na+ ions decreases in the ﬁlter, the current carried by them decreases simultaneously. The main difference between Fig. 8 and Fig. 6 is that the curves for different combinations of H and R are farther apart when ﬁlter volume is changed (Fig. 6) than in the case when it is unchanged (Fig. 8). We conclude that changing the pore shape has less effect if the pore volume is conserved. Thus, pore shape (at constant volume) provides an additional engineering variable with which we can ﬁnetune the selectivity properties of the model. Pore radius can be especially important in the competition between ions of different size, in particular for the sodium channel, where Na+ versus K+ selectivity seems to arise from a deeper depletion zone for K+ than for Na+ [23]. 3.3. Softness of oxygen conﬁnement does not inﬂuence selectivity

Fig. 8. (a) Normalized conductances as a function of log10[CaCl2] in M added to a 30-mM NaCl background for channels with varying ﬁlter lengths and radii while keeping ﬁlter volume constant (ɛpr = 10). The times symbols show the experimental results of Almers et al. [2,3]. (b) Normalized ionic conductances of the pore for Na+ and Ca2+ as given by Eq. 4. (c) The average number of Na2+ and Ca2+ ions in the selectivity ﬁlter. In panels b and c, increasing curves refer to Ca2+, while decreasing curves refer to Na+.

Our results so far imply that particle density in the ﬁlter and packing effects resulting in depletion zones are important determinants of selectivity. Packing of oxygen ions is strongly inﬂuenced by the way we conﬁne them. In our earlier studies (and in the simulations shown so far in this work), the oxygen was conﬁned by hard walls at ±H/2 (corresponding to a → ∞ in Eq. 1). This hard wall conﬁnement results in an artiﬁcially high peak of oxygen at these

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resolution structure (which is not available for Ca2+ channels), we model the bulk of the protein in an abstract, low-resolution structure. This structure is fully speciﬁed by a small number of parameters whose physical meaning is self-evident. Our simulations showed that ionic density is an important determinant of Ca2+ versus Na+ selectivity in calcium channels. If the number of structural ions in the ﬁlter (8 O1/2− ions in this study) and polarization properties (dielectric coefﬁcient of the protein) are ﬁxed, it is the volume that determines the density. Simulations with different pore shapes with and without conserved volume showed that variation in selectivity is larger if the volume is changed. Channel shape at ﬁxed volume is also important, but its inﬂuence is smaller. Strong variations of particle packing have surprisingly small signiﬁcance for selective (relative) conduction, but they do have interesting consequences regarding the absolute value of conductance (for example, the size of the Na+ leak at physiological [Ca2+]). Finally, we have shown that conﬁning the oxygen ions by soft walls instead of hard walls has little effect on selectivity. Fig. 10. Normalized conductances as a function of log10[CaCl2] in M added to a 30-mM NaCl background for values of the “softness” parameter a (channel parameters are R = 3.5 Å, H = 10 Å and ɛpr = 10). The × symbols show the experimental results of Almers et al. [2,3].

walls which is a well-known hard sphere packing effect (see, for example, insets of Fig. 5). In real selectivity ﬁlters, made by proteins, conﬁnement is unlikely to be so hard. Therefore, we examine the role of the hard wall. Does the hard wall produce artifactual depletion zones of Ca2+ or Na+? Does it have any artiﬁcial effect that would seriously inﬂuence our conclusions drawn from this channel model? We performed simulations with different values of the a parameter of the soft conﬁning potential given in Eq. 1 for the geometry H = 10 Å and R = 3.5 Å. As expected, the distribution of O1/2− ions is signiﬁcantly inﬂuenced by the way they are conﬁned as shown in the top panel of Fig. 9 for [CaCl2] = 10− 6 M. The oxygen ions distribute over the ﬁlter more evenly when a softer conﬁnement allows them more ﬂexibility at the ﬁlter entrances; the high peaks are suppressed. Remarkably, the proﬁles of Na+ and Ca2+ are much less inﬂuenced. The conductance of the channel is the same whether the channel is constructed with hard or soft walls (Fig. 10) even though the resulting distribution of the side chains is quite different. This is a satisfying result because it supports our hypothesis: it is the conﬁnement provided by the protein and the density of the charges in the selectivity ﬁlter that really matter, and as long as those coarse features are maintained, then smaller details of protein structure have minor effects on selectivity. 4. Summary Our reduced model of the selectivity ﬁlter is not intended to be a realistic structural representation of the calcium channel. Instead, it is used to study the role of fundamental classical physical interactions (electrostatics and hard sphere exclusion) and how these terms create selectivity to ﬁrst order. The success of the model and the CSC mechanism implies that these terms should be present in more detailed models of the calcium channel. Molecular dynamics simulations of atomistic models of channels have been chieﬂy concentrated on potassium channels whose structure is known [51–55]. Simulations of assumed atomic structures of calcium channels using that of the potassium channel as a template have also been published [56– 58]. Neither of these simulations, nevertheless, relate their results to experimental conductivities as functions of varying bath compositions as we do. Atomic structure, nevertheless, has a close relation to the parameters of our reduced model. The polarization properties of the protein surrounding the pore (ɛpr in our model), as well as the length H and radius R of the selectivity ﬁlter are all structural parameters that are essentially determined by protein structure. In place of high-

Acknowledgment The authors thank the Ira and Marylou Fulton Supercomputing Center at BYU, the Hungarian National Research Fund (OTKA K75132, to BD) and NIH (GM076013 to BE).

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